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Let $G$ be a connected reductive group over a field $k$. Let $E$ be a $G$-bundle, then we can form the adjoint bundle $ad(E)$ which is a Lie algebra bundle over $k$. As a vector bundle it is trivial, but is there a criterion to know when it is trivial as a Lie algebra bundle?

More generally the same question could be asked over a base scheme $S$ and for a $G$-torsor $E$ such that $ad(E)$ is a trivial vector bundle over $S$.

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